A NEW APPROACH TO THE ANALYSIS OF FREE ROTATIONS OF RIGID BODIES

Later POINSOT offered the famous geometrical interpretation to show re al rotation of the body. In modern literature this problem is called t he case of the integrability by Euler-Poinsot. Up to Mow no theoretica l adaptations were made to the classical solution, the description of which is presented in all books on dynamics of rigid bodies. The class ical solution allows perfectly to find the rotations, i.e. angular vel ocities. of tile body. However, the determination of the turns, i.e. a ngles, does nor impress so much. Moreover, it may be shown that the ap plication of the Eulerian angles to this problem is nor the best way b ecause of several reasons. Firstly, the Euler inn angles, as a rule, g ive a representation which is rather difficult for interpretation. Sec ondly, this representation generates difficulties for the numerical re alization on computers. By these reasons ii seems to be useful to give an alternative approach to the analysis of the Euler-Poinsot problem, that is based on the concept of the tensor of turn called in the sequ el turn-tensor. Some main facts concerning the turn-tensor are present ed in the introduction, where the new theorem on the representation of the turn-tensor is given. The theorem allows to simplify the solution of problems of the dynamics of rigid bodies. In Euler-Poinsot's probl em it is not difficult to find four first integrals of the basic equat ions. Three of them are well known. They express that the angular mome ntum vector of the body is constant. The fourth integral is that of en ergy which is directly expressed in terms of turns rather than of angu lar velocities. The energy integral in such a form allows to construct the most suitable representation of the turn-tensor to make the pictu re of turns of the body clear. It is found that there exist three and only three different types of rotations. Two of them give stable rotat ions, and the third type describes an unstable rotation. The type of r otation is determined for the given body by initial conditions only. I n fact, the third type of rotation is the separatrix between two stabl e types of rotations. Under some conditions the stable rotations at ce rtain moments of time can be very close to each other. Thus it is poss ible for the body to jump from one stable solution to another stable r otation. For example, the body can be rotating around the axis with mi nimal moment of inertia and there upon it can change the rotation to b egin the rotation around the axis with maximal moment of inertia. Of c ourse, small perturbations acting on the body are needed to provoke su ch a situation. As final result the problem is reduced to the integrat ion of the simple differential equation of the first order, the soluti on of which is a monotonically increasing function. All required quant ities can be expressed in terms of this function. It is shown how to s ee the turns of the body without integration of the equation if initia l conditions are given.